Estimates for the Number of Sums and Products and for Exponential Sums in Fields of Prime Order

By Jean Bourgain, A. Glibichuk, Sergei V. Konyagin. 2006

239 citations recorded by the source corpus.

Subfields: Algebra and Number Theory, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/cab10afa5aee8e283d91e43db1d108b227c3f42c

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Citations in the corpus

NEW RESULTS ON SUM‐PRODUCT TYPE GROWTH OVER FIELDS (2017, unknown): “1]; see also [22, 7].”
A slight improvement to Garaev’s sum product estimate (2007, background): “It is fun (and useful) to prove lower bounds on max(|A+A|, |AA|) (see e.g. [BKT],[ BGK ],[G]).”
On a variant of sum-product estimates and explicit exponential sum bounds in prime fields (2009, background): “From [7, 9] it is known that sum-product results imply new exponential sum estimates in Fp.”
Expansion in $SL_d(O_K/I)$, $I$ square-free (2010, background): “However the contradiction could also be decuded from the results of [10] or [9].”
Growth and generation in $\mathrm{SL}_2(\mathbb{Z}/p \mathbb{Z})$ (2008, unknown): “Proof. Let us take Fourier transforms and proceed as in the beginning of the proof of Thm. 6 in [ BGK ]:”
Multilinear Exponential Sums in Prime Fields Under Optimal Entropy Condition on the Sources (2009, background): “It provides the expected generalization of the well-known inequality for r = 2. We also establish a new result on Gauss sums for multiplicative subgroups H of F �, obtaining a nontrivial estimate provided log |H| > C log p log log p . This is a further improvement on [ BGK ].”
Approximate Subgroups of Linear Groups (2010, methodology): “56], and is essentially contained in [9, 10].”
Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture (2007, background): “See also [2], [4], [8], [7], [11], [14], [19], [20] and the references contained therein on recent progress related to this problem and its analogs.”